AUBURN MECH 7220 - Laminar External Flow

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Laminar External FlowThe boundary layerMomentum transfer on a flat plateOrder--of--magnitude analysisSimilarity solutionHomework Exercises5 Laminar External Flow5.1 The boundary layerA boundary layer is an easy physical concept to grasp; it is the result of the no–slip boundary conditionas fluid flows over the surface of an object. Because of viscosity, the fluid directly adjacent to the object issheared, and this effect is confined to a relatively thin layer parallel to the surface, i.e., the boundary layer.A boundary layer is also a mathematical concept, and ”boundary layer” behavior can occur wheneverthe higher–order derivatives in a differential equation are multiplied by relatively small parameters. Thisstatement is more arcane, yet it is important in terms of gaining insight into why physical boundary layersare ”thin”, and what solution methods can be used to solve for the transport of momentum and heat acrossthe boundary layers.A simple example – both physical and mathematical – of boundary layer behavior can be seen in a 1–Dconvective–diffusive problem. Say a uniform, 1–D flow enters a control volume with temperature T0. Atsome distance L downstream, the flow is brought to a new temperature T1. All the while the density ρ andthe velocity u stay constant. The boundary at x = L might represent, for example, a porous surface whichis maintained at T1.The steady 1–D energy equation for this situation isρUCdTdx= kd2Tdx2(1)Denote non–dimensional variables asT =T − T1T0− T1, x =xL(2)and the DE becomesdTdx=1P eLd2Tdx2(3)T (0) = 1, T (1) = 0 (4)withP eL=u Lα=ρ C u Lk(5)The solution to this problem isT (x, P eL) =eP eL− eP eLxeP eL− 1(6)A plot of the results is shown below.0 0.2 0.4 0.6 0.8 1x/L00.20.40.60.81(T-T1)/(T0-T1)PeL=0.1PeL=1 PeL=10PeL=100Note that as P eLincreases, the temperature distribution goes from being linear to increasingly ”piled up”into a narrow region adjacent to the x = L boundary. Indeed, for P eL= 100 the temperature is basically1uniform at T = 1 – except for a small region of steep gradient at the outflow surface. This region of steepgradient would be considered a boundary layer. The 2nd–order derivative, for this case, is multipled bythe small parameter 1/100 = 0. 01. This might lead one to assume that the 2ndderivative term could beneglected from the DE – and it could, through most of the flow region. Getting rid of this term would resultindTdx= 0 −→ T = constant = 1 (7)where the constant is evaluated from the first BC. However, the problem has two boundary conditions, and thezero–temperature condition must be maintained at the outflow wall. A first–order DE cannot accommodatetwo boundary conditions, so somewhere the second–order derivative term must become significant. It willbecome significant in the neighborhood of the T = 0 surface.The thickness of the boundary layer, for this simple problem, can be determined by reformulating theproblem. Reverse the coordinate direction, so that x runs from the T = 0 surface, and define a scaled (orstretched) coordinate ˜x via˜x = x P eL(8)The scaled DE becomes−dTd˜x=d2Td˜x2(9)T (0) = 0, T (˜x → ∞) → 1 (10)In this view the far surface (with T = 1) now exists at ˜x → ∞: the model describes only the behavior withinthe boundary layer. The solution is nowT (˜x) = 1 − e−˜x(11)and this indicates that the temperature reaches T = 0.99 for ˜x ≈ 4.61. The thickness of the boundary layeris thereforeδbl≈4.61 LP eL=4.61 αu(12)Some points to make regarding the boundary layer effect are:1. The boundary layer represents the region in which all derivatives in the governing DE contribute signif-icantly to the transport process. Outside of this region transport is dominated by a single mechanism(convection, in this case).2. The details of the boundary layer (i.e., the thickness) can be neglected if one is interested in predictingonly the bulk characteristics of the flow. For the simple model examined here, the bulk (or freestream)characteristics would be predicted by Eq. (7), and this result gives the conditions at the edge of theboundary layer.3. On the other hand, the details of the boundary layer are absolutely critical if one wishes to predict therate of transport to/from a surface that is exposed to a flow. Note that Eq. (7) cannot predict the rateof heat transfer to the T = 0 surface.4. On an order–of–magnitude level, the rate of transport across the boundary layer will beq00≈ kT∞− Tsδbl(13)This approximation does not have much relevance to the simple 1–D model, because the heat flux mustbe q00= ρ u C(T1− T0) by virtue of the first law for the system (and this is precisely what the aboveequation would give). It will have more bearing in 2–D flow situations.25.2 Momentum transfer on a flat plateConsider a situation in which a flat plate is exposed to a flow that runs parallel to its surface. Far from thesurface of the plate the velocity is u∞. A boundary layer will form at the leading edge of the plate, andthe thickness of the layer will grow with distance downstream. Everywhere the flow remains laminar (theconditions for laminar BL flow will be discussed below). The objective is to predict the velocity profile inthe boundary layer.Assuming incompressible flow, the continuity and momentum equations for the situation are∂u∂x+∂v∂y= 0 (14)u∂u∂x+ v∂u∂y= −1ρdPdx+ ν∂2u∂y2(15)where x, y are the streamwise and normal coordinates, and u, v are the associated components of velocity.Assumptions are 1) there is negligible pressure gradient in the normal direction, so that P = P (x), 2)streamwise diffusion of momentum is negligible relative to streamwise convection.Outside the boundary layer the flow is 1–D in the x direction, so u = u∞= constant and v = 0. Themomentum equation, applied in this region, would show that dP/dx = 0. The pressure gradient term willbe neglected from here on out, yet it will become relevant (i.e., non–zero) whenever the external flow isaccelerating or decelerating, as would be the case for a surface that is tilted with respect to the flow.5.2.1 Order–of–magnitude analysisIt is possible to use Eqs. (14) and (15) to estimate how the boundary layer thickness changes with position xfrom the leading edge. At some point x, at which the boundary layer is δ thick, the change in u is, at most,u∞over the distance x. The normal component of velocity goes from 0 to some characteristic


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